Question

If $A,B,C$ are angles of an acute angle triangle. Then minimum value of
$\left|\begin{array}{ccc}(tanB+tanC{)}^2& ta{n}^{2}A& ta{n}^{2}A\\ ta{n}^{2}B& (tanC+tanA{)}^2& ta{n}^{2}B\\ ta{n}^{2}C& ta{n}^{2}C& (tanA+tanB{)}^2\end{array}\right|$ is

Answer 1

$LettanA=a,tanB=b,tanC=c\left|\begin{array}{ccc}(b+c{)}^2& a^2& a^2\\ b^2& (c+a{)}^2& b^2\\ c^2& c^2& (a+b{)}^2\end{array}\right|$
$C_2\to C_2-C_1,C_3\to C_3-C_1$
$=(a+b+c{)}^2\left|\begin{array}{ccc}(b+c{)}^2& a-b-c& a-b-c\\ b^2& c+a-b& 0\\ c^2& 0& a+b-c\end{array}\right|$
$R_1\to R_1-(R_2+R_3)$
$=(a+b+c{)}^2\left|\begin{array}{ccc}2bc& -2c& -2b\\ b^2& c+a-b& 0\\ c^2& 0& a+b-c\end{array}\right|$
$C_2\to C_2+\frac{1}{b}{C}_1,C_3\to C_3+\frac{1}{c}{C}_1$
$=(a+b+c{)}^2\left|\begin{array}{ccc}2bc& 0& 0\\ b^2& c+a& \frac{b^2}{c}\\ c^2& \frac{c^2}{b}& a+b\end{array}\right|=(a+b+c{)}^2\times 2bc\left[\right(a+c\left)\right(a+b)-bc]$
$=2abc(a+b+c{)}^3$
$For\Delta ABC,weknowtanA+tanB+tanC=tanAtanBtanC\Rightarrow a+b+c=abc$
$\therefore Givenexpression=2(abc{)}^4$
The product of three non-zero numbers a, b, c is minimum, when all are equal.
$\therefore \tan A=\tan B=\tan C=\sqrt{3}$
$\text{Minimum value of the given expression is}(=2(\sqrt{3}\times \sqrt{3}\times \sqrt{3}{)}^4=2\times 3^4\times 3^2=2\times 729=1458$